Öz
Let $\mathbb{A}$ be the affine group, $\Phi_1, \Phi_2$ be Young functions. We study the Orlicz amalgam spaces $W(L^{\Phi_1} (\mathbb{A}),L^{\Phi_2} (\mathbb{A}))$ defined on $\mathbb{A}$, where the local and global component spaces are the Orlicz spaces $L^{\Phi_1}(\mathbb{A})$ and $L^{\Phi_2}(\mathbb{A})$, respectively. We obtain an equivalent discrete norm on the amalgam space $W(L^{\Phi_1} (\mathbb{A}), L^{\Phi_2} (\mathbb{A}))$ using the constructions related to the affine group. Using the discrete norm we compute the dual space of $W(L^{\Phi_1} (\mathbb{A}),L^{\Phi_2} (\mathbb{A}))$. We also prove that the Orlicz amalgam space is a left $L^1(\mathbb{A})$-module with respect to convolution under certain conditions. Finally, we investigate some inclusion relations between the Orlicz amalgam spaces.
Anahtar Kelimeler
Amalgam spaces Orlicz spaces affine group discrete norms convolution
Teşekkür
I would like to thank Prof. S. Öztop for critical reading of the manuscript and helpful suggestions on the subject.
Kaynakça
- [1] B. Ars and S. Öztop, Wiener amalgam spaces with respect to Orlicz spaces on the affine group, J. Pseudo. Differ. Oper. Appl. 14, 23, 2023.
- [2] A. Benedek and R. Panzone, The spaces $L^p$ with mixed norm, Duke Math. J. 28, 301-324, 1961.
- [3] J.P. Bertrandias, C. Datry and C. Dupuis, Unions et intersections despaces Lp invariantes par translation ou convolution, Ann. Inst. Fourier 28, 53-84, 1978.
- [4] R.C. Busby and H.A. Smith, Product-convolution operators and mixed-norm spaces, Trans. Amer. Math. Soc. 263, 309-341, 1981.
- [5] D.L. Cohn, Measure Theory, 2nd ed., Birkhäuser/Springer, New York, 2013.
- [6] E. Cordero and F. Nicola, Sharpness of some properties of Wiener amalgam and modulation spaces, Bull. Aust. Math. Soc. 80, 105-116, 2009.
- [7] H.G. Feichtinger, Banach convolution algebras of Wiener type, in: Functions, Series, Operators, Colloq. Math. Soc. 35, 509-524, János Bolyai North-Holland, Amsterdam, 1983.
- [8] H.G. Feichtinger, Banach spaces of distributions defined by decomposition methods, II, Math. Nachr. 132, 207237, 1987.
- [9] H.G. Feichtinger and P. Gröbner, Banach spaces of distributions defined by decomposition methods, I, Math. Nachr. 123, 97-120, 1985.
- [10] H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal., 86, 307-340, 1989.
- [11] C. Heil, An introduction to weighted Wiener amalgams, in: Wavelets and Their Applications, 183216, Allied Publishers, New Delhi, 2003.
- [12] C. Heil and G. Kutyniok, The homogeneous approximation property for wavelet frames, J. Approx. Theory, 147, 28-46, 2007.
- [13] C. Heil and G. Kutyniok, Convolution and Wiener amalgam spaces on the affine group, in: Recent Advances in Computational Science, 209217, World Scientific, Singapore, 2008.
- [14] F. Holland, Harmonic analysis on amalgams of $L^p$ and $\ell^q$, J. London Math. Soc. 10, 295-305, 1975.
- [15] A. Osançlıol and S. Öztop, Weighted Orlicz algebras on locally compact groups, J. Aust. Math. Soc. 99, 399-414, 2015.
- [16] S. Öztop and E. Samei, Twisted Orlicz algebras I, Studia Mathematica, 236, 271-296, 2017.
- [17] M.M. Rao, Extensions of the Hausdorff-Young theorem, Israel J. Math. 6, 133-149, 1967.
- [18] M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
- [19] M.M. Rao and Z.D. Ren, Application of Orlicz Spaces, Marcel Dekker, New York, 2002.
- [20] M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, in: Evolution Equations of Hyperbolic and Schrödinger Type, Progr. Math., 301, 267283, Birkhäuser/Springer, 2012.
- [21] N. Wiener, On the representation of functions by trigonometric integrals, Math. Z., 24, 575616, 1926.
- [22] N. Wiener, Tauberian theorems, Ann. of Math., 33, 1100, 1932.
- [23] N. Wiener, The Fourier Integral and Certain of its Applications, Cambridge Univ. Press, Cambridge, 1933.
Öz
Kaynakça
- [1] B. Ars and S. Öztop, Wiener amalgam spaces with respect to Orlicz spaces on the affine group, J. Pseudo. Differ. Oper. Appl. 14, 23, 2023.
- [2] A. Benedek and R. Panzone, The spaces $L^p$ with mixed norm, Duke Math. J. 28, 301-324, 1961.
- [3] J.P. Bertrandias, C. Datry and C. Dupuis, Unions et intersections despaces Lp invariantes par translation ou convolution, Ann. Inst. Fourier 28, 53-84, 1978.
- [4] R.C. Busby and H.A. Smith, Product-convolution operators and mixed-norm spaces, Trans. Amer. Math. Soc. 263, 309-341, 1981.
- [5] D.L. Cohn, Measure Theory, 2nd ed., Birkhäuser/Springer, New York, 2013.
- [6] E. Cordero and F. Nicola, Sharpness of some properties of Wiener amalgam and modulation spaces, Bull. Aust. Math. Soc. 80, 105-116, 2009.
- [7] H.G. Feichtinger, Banach convolution algebras of Wiener type, in: Functions, Series, Operators, Colloq. Math. Soc. 35, 509-524, János Bolyai North-Holland, Amsterdam, 1983.
- [8] H.G. Feichtinger, Banach spaces of distributions defined by decomposition methods, II, Math. Nachr. 132, 207237, 1987.
- [9] H.G. Feichtinger and P. Gröbner, Banach spaces of distributions defined by decomposition methods, I, Math. Nachr. 123, 97-120, 1985.
- [10] H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal., 86, 307-340, 1989.
- [11] C. Heil, An introduction to weighted Wiener amalgams, in: Wavelets and Their Applications, 183216, Allied Publishers, New Delhi, 2003.
- [12] C. Heil and G. Kutyniok, The homogeneous approximation property for wavelet frames, J. Approx. Theory, 147, 28-46, 2007.
- [13] C. Heil and G. Kutyniok, Convolution and Wiener amalgam spaces on the affine group, in: Recent Advances in Computational Science, 209217, World Scientific, Singapore, 2008.
- [14] F. Holland, Harmonic analysis on amalgams of $L^p$ and $\ell^q$, J. London Math. Soc. 10, 295-305, 1975.
- [15] A. Osançlıol and S. Öztop, Weighted Orlicz algebras on locally compact groups, J. Aust. Math. Soc. 99, 399-414, 2015.
- [16] S. Öztop and E. Samei, Twisted Orlicz algebras I, Studia Mathematica, 236, 271-296, 2017.
- [17] M.M. Rao, Extensions of the Hausdorff-Young theorem, Israel J. Math. 6, 133-149, 1967.
- [18] M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
- [19] M.M. Rao and Z.D. Ren, Application of Orlicz Spaces, Marcel Dekker, New York, 2002.
- [20] M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, in: Evolution Equations of Hyperbolic and Schrödinger Type, Progr. Math., 301, 267283, Birkhäuser/Springer, 2012.
- [21] N. Wiener, On the representation of functions by trigonometric integrals, Math. Z., 24, 575616, 1926.
- [22] N. Wiener, Tauberian theorems, Ann. of Math., 33, 1100, 1932.
- [23] N. Wiener, The Fourier Integral and Certain of its Applications, Cambridge Univ. Press, Cambridge, 1933.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Lie Grupları, Harmonik ve Fourier Analizi, Operatör Cebirleri ve Fonksiyonel Analiz |
| Bölüm | Matematik |
| Yazarlar | |
| Erken Görünüm Tarihi | 27 Ağustos 2024 |
| Yayımlanma Tarihi | 28 Nisan 2025 |
| Gönderilme Tarihi | 29 Mart 2024 |
| Kabul Tarihi | 1 Haziran 2024 |
| Yayımlandığı Sayı | Yıl 2025 Cilt: 54 Sayı: 2 |